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Expanding triple (cross) product

  • #1
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Homework Statement



Use (the bac-cab rule) to expand this triple product:L = mr x (ω x r)

If r is perpendicular to ω, show that you obtain the elementary formula, angular momentum = mvr.

(The bold letters are vectors.)

Homework Equations



A X (B X C) = (A[itex]\cdot[/itex]C)B - (A[itex]\cdot[/itex]B)C

The Attempt at a Solution



Well, simply doing the bac-cab rule, I get

(mr [itex]\cdot[/itex]ω)r - (mr [itex]\cdot[/itex]r)ω

Which isn't even close to the answer in the book, which is:

L=m[r2ω-(ω [itex]\cdot[/itex]r)r]

No idea how they got that.

Even if I distribute the r, they don't have it distributed to the ω, and on the right term, they don't distribute the ω. I don't understand.
Thanks
 

Answers and Replies

  • #2

The Attempt at a Solution



Well, simply doing the bac-cab rule, I get

(mr [itex]\cdot[/itex]ω)r - (mr [itex]\cdot[/itex]r)ω
This is wrong, going by the rule we get
m[(r.r)ω-(r.ω)r]
On simplification we'll get the answer.....
Regards
 
  • #3
575
47
This is wrong, going by the rule we get
m[(r.r)ω-(r.ω)r]
On simplification we'll get the answer.....
Regards
OK yeah I lost track of which one was A, B, and C.

But this is what I get:
(mr[itex]\cdot[/itex]r)ω-(mr[itex]\cdot[/itex]ω)r

r[itex]\cdot[/itex]r simplifies to r2 because it's the vector dotted into itself, which produces the magnitude of that vector squared?

And according to the solution, why didn't they distribute the r through the parentheses in the second term?

Thanks.
 
  • #4
vela
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What specifically do you mean by "distribute the r through the parentheses"?
 
  • #5
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What specifically do you mean by "distribute the r through the parentheses"?
Oh ok, so that r isn't allowed to go into the parentheses until the r is dotted into the ω?
 
  • #6
vela
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I guess I still don't know what you're trying to do. ##\vec{\omega}\cdot\vec{r}## is a scalar, and the result of the product multiplies ##\vec{r}##. By pulling ##\vec{r}## into the parentheses, what do you intend to accomplish?
 

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